Polynomial identity rings

the long version

A ring R satisfies a polynomial identity if there is a polynomial f in noncommuting variables which vanishes under substitutions from R. For example, commutative rings satisfy the polynomial f(x,y) = xy - yx and exterior algebras satisfy the polynomial f(x,y,z) = (xy - yx)z - z(xy - yx). "Satisfying a polynomial identity" is often regarded as a generalization of commutativity. These lecture notes treat polynomial identity rings from both the combinatorial and structural points of view. The former studies the ideal of polynomial identities satisfied by a ring R. The latter studies the properties of rings which satisfy a polynomial identity. The greater part of recent research in polynomial identity rings is about combinatorial questions, and the combinatorial part of the lecture notes gives an up-to-date account of recent research. On the other hand, the main structural results have been known for some time, and the emphasis there is on a presentation accessible to newcomers to the subject. The intended audience is graduate students in algebra, and researchers in algebra, combinatorics and invariant theory.